Schmidt and Other Multipartite Entanglement Measures of Graph States

Graph states play an important role in quantum information theory through their connection to measurement-based computing, error correcting codes, secret sharing, and stabilizer computation. While much effort has gone into the entanglement properties of such states, primarily bicolorable graphs have been characterized. In this work we prove various multipartite entanglement properties for odd cycle graphs. We first start by tightening the bounds on Schmidt the measure of such states to (n, n+log3]. This improves previous bounds on the entanglement cost for creating odd-cycle graph states using local operations and classical communication (LOCC) with shared entanglement. Next, we prove that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states: either 0 or near maximal based solely on if the graph is connected. Lastly, we show that the n-tangle, which is related to stochastic LOCC invariance, can be computed in a graph theoretic manner: it is one if all vertices have odd degree, and zero otherwise. These dichotomous results indicate that other entanglement measures may be more insightful for graph states.